Formula for Area of Segment of a Circle with Steps and Examples
The concept of area of segment of a circle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Area of Segment of a Circle?
A segment of a circle is defined as the region enclosed by a chord and the corresponding arc lying between the chord’s endpoints. You’ll find this concept applied in areas such as finding pizza slice areas, designing round gardens, and calculating segments in engineering drawings.
Key Formula for Area of Segment of a Circle
Here’s the standard formula: \( \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \)
In radians, the formula commonly used is: \( \text{Area} = \dfrac{1}{2} r^2 (\theta - \sin\theta) \) where \( r \) is the radius and \( \theta \) is the angle in radians.
In degrees, substitute \( \theta \) in radians as \( \theta = \dfrac{\pi}{180} \times \text{angle in degrees} \).
Cross-Disciplinary Usage
Area of segment of a circle is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, such as calculating fields, parts of wheels, or circular data segments.
Step-by-Step Illustration
Let’s see an example: Find the area of a segment of a circle with radius 10 cm and a central angle of 60°.
\( 60^\circ = \dfrac{\pi}{3} \) radians
2. Use the segment formula:
\( \text{Area} = \dfrac{1}{2} \times 10^2 \times \left( \dfrac{\pi}{3} - \sin \dfrac{\pi}{3} \right) \)
\( \sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2} \approx 0.866 \)
\( \text{Area} = 50 \times \left( 1.047 - 0.866 \right) = 50 \times 0.181 = 9.05 \) cm²
Final Answer: The segment area is about 9.05 cm²
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for the area of minor segment in exams: If the question uses a small angle (like 30°, 45°, 60°), remember that \(\sin\theta\) values are standard and easy to recall. Substitute these directly for fast calculation in the formula. This helps speed up MCQs and time-bound tests.
- For 60°, \(\sin 60° = 0.866\), for 30°, \(\sin 30° = 0.5\), for 45°, \(\sin 45° = 0.707\).
- Plug directly into the formula: \( \text{Area} = \frac{1}{2} r^2 (\theta - \sin\theta) \) where \( \theta \) is in radians.
Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.
Try These Yourself
- Find the area of the segment of a circle with radius 5 cm and angle 90°.
- If a chord is 8 cm long in a circle of radius 5 cm, find the area of the minor segment (use formula with chord length).
- Calculate the area of the major segment if the central angle is 120° and radius is 7 cm.
- Find the area of a semicircular segment (hint: angle is 180°).
Frequent Errors and Misunderstandings
- Mixing up sector and segment area formulas.
- Forgetting to convert degrees to radians before using the standard formula.
- Confusing minor segment (smaller part) with major segment (larger part).
- Missing units in the final answer (cm² or m²).
- Not subtracting the triangle area correctly when using the sector minus triangle formula.
Relation to Other Concepts
The idea of area of segment of a circle connects closely with topics such as Area of a Circle and Area of Sector. Mastering this helps with understanding more advanced chapters in Mensuration and Geometry, especially in classes 9, 10, and beyond.
Classroom Tip
A quick way to remember the area of segment formula is: “Sector area minus triangle area.” Build the diagram, shade the segment, and always double-check if you need the minor or major segment. Vedantu’s teachers often use circle cut-outs and colored sections to help students visualize and retain these ideas easily in live classes.
We explored area of segment of a circle—from definition, formula, example calculation, mistakes, and how it connects to other subjects. Continue practicing with Vedantu to become confident in solving problems using this important Maths concept.
Keep Learning: Internal Links
- Area of a Circle — Understand circles before segments.
- Chord of a Circle — Explore how chords relate to segments.
- Area of Sector and Segment of a Circle — For combined exam sum practice.
- Arc of a Circle — Learn about arcs, essential in segment area problems.
- Properties of Circle — Master all-round circle concepts.
FAQs on Area of a Circle Segment Explained Clearly
1. What is the area of a segment of a circle?
The area of a segment of a circle is the region bounded by a chord and the corresponding arc, and it is found by subtracting the area of the triangle from the area of the sector. The formula is:
Area of segment = Area of sector − Area of triangle
For a central angle θ (in degrees) and radius r:
- Area of sector = (θ/360) × πr²
- Area of triangle = (1/2)r² sinθ
2. What is the formula for the area of a segment in radians?
The area of a segment (in radians) is given by (1/2)r²(θ − sinθ), where θ is in radians. This formula comes from:
- Area of sector = (1/2)r²θ
- Area of triangle = (1/2)r² sinθ
Area of segment = (1/2)r²θ − (1/2)r² sinθ = (1/2)r²(θ − sinθ). This is the most compact and commonly used formula in higher mathematics.
3. How do you find the area of a segment step by step?
To find the area of a segment, subtract the area of the triangle from the area of the sector.
Follow these steps:
- Step 1: Find Area of sector = (θ/360) × πr² (if θ is in degrees).
- Step 2: Find Area of triangle = (1/2)r² sinθ.
- Step 3: Subtract: Segment area = Sector − Triangle.
- Sector = (60/360) × π × 49 = 25.67 cm²
- Triangle = (1/2) × 49 × sin60° ≈ 21.22 cm²
- Segment ≈ 4.45 cm²
4. What is the difference between a sector and a segment of a circle?
The sector is the region between two radii and an arc, while the segment is the region between a chord and an arc.
Key differences:
- A sector looks like a slice of pizza.
- A segment is formed when a chord cuts off part of a circle.
- Segment area = Sector − Triangle.
5. What are the types of segments in a circle?
A circle has two types of segments: minor segment and major segment.
- The minor segment is the smaller region cut off by a chord.
- The major segment is the larger remaining part of the circle.
6. Can you give an example of finding the area of a circular segment?
Yes, the area of a circular segment can be calculated using the sector-minus-triangle method.
Example: If r = 5 cm and θ = 90°:
- Sector = (90/360) × π × 25 = 19.63 cm²
- Triangle = (1/2) × 25 × sin90° = 12.5 cm²
- Segment = 19.63 − 12.5 = 7.13 cm²
7. How do you find the area of a segment when the chord length is given?
To find the area of a segment from chord length, first determine the central angle using trigonometry, then apply the segment formula.
Steps:
- Use c = 2r sin(θ/2) to find θ.
- Convert θ into radians if needed.
- Apply Area = (1/2)r²(θ − sinθ).
8. Why do we subtract the triangle area from the sector area?
We subtract the triangle area because the segment is only the curved region between the chord and arc, not the entire sector.
The sector includes:
- The curved arc region
- The triangular region formed by two radii
9. What is the area of a semicircular segment?
The area of a semicircular segment (θ = 180°) equals half the circle minus the area of the triangle formed by the diameter.
Since sin180° = 0:
- Segment area = (1/2)πr² − 0
10. What are common mistakes when calculating the area of a segment?
The most common mistake when finding the area of a segment is mixing degrees and radians in formulas.
Other frequent errors include:
- Using θ in degrees in the formula (1/2)r²(θ − sinθ) without converting to radians.
- Forgetting to subtract the triangle area.
- Using incorrect sine values.
- Confusing a sector with a segment.
